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We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible shear. Our treatment was motivated by the problem of the simulation of catheter navigation in a network of blood vessels, where this description is directly useful. In this context, we derive the Euler differential equations that characterize equilibrium configurations of stretch-free thin rods. We apply perturbation methods, used in time-dependent quantum theory, to the thin rod equations to describe incremental deformations of partially constrained rods. Further, our formalism leads naturally to a new and efficient finite element method valid for arbitrary deformations of thin rods with negligible stretch. Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.

The storage of biased patterns is examined in neural networks with sign-constrained synapses. Every neuron has outgoing synapses which are either all inhibitory or all excitatory. For random patterns stored in such networks, it is known that the presence of a discrete gauge symmetry makes the maximal storage capacity independent of the proportion of excitatory neurons to inhibitory neurons. When the stored patterns are biased, however, this discrete gauge symmetry is broken, with the result that the maximal capacity depends on the proportion of excitatory neurons to inhibitory ones. The dependence of the capacity on the fraction of excitatory neurons in the network, f, is calculated using the space of interactions approach. It is found that the storage capacity is maximal at f=0.5; this result is true regardless of the particular value of the bias in the stored patterns. The significance of this result in the neurophysiological context is discussed.

A neural network model with bipolar synaptic couplings is studied with a global constraint on the synapses which earn neuron receives, corresponding to a bias of the couplings. When the network is used to store biased patterns, it is shown that the maximal storage capacity for any non-zero bias of the stored patterns corresponds to a particular type of the connectivity pattern of the network as encoded in the constraint. The 'space of interactions' approach yields the dependence of the storage capacity on the distribution of the couplings for every pattern bias. It turns out that the optimal coupling bias is non-zero and independent of the bias in the patterns.